Question

the area of one of the small right triangles outlined in blue is a cm², while the area of the square outlined in red is b cm². which expressions show the area of the shaded region in terms of a and b? check all that apply. 4a + b 8a + b 2a + b 1.5b 12a

Explanation:

Step1: Analyze the figure's components

The red square (area \( B \)) is made up of 4 small right triangles (each with area \( A \))? Wait, no. Wait, the blue triangles: let's count the number of small right triangles (area \( A \)) in the shaded region. Wait, looking at the figure, the red square (area \( B \)) – let's see, how many triangles make up the square? A square can be divided into 4 right triangles (if it's a square with diagonals), but here, maybe the square \( B \) is composed of 4 triangles? Wait, no, the problem says the small right triangle is \( A \). Let's re-examine: the shaded region – how many small triangles (area \( A \)) and the square \( B \)? Wait, maybe the square \( B \) is made of 4 triangles, but the shaded area has some triangles and the square? Wait, no, the options include \( 4A + B \), \( 8A + B \), etc. Wait, maybe the square \( B \) is composed of 4 triangles (each \( A \)), so \( B = 4A \)? Wait, no, the problem says "the area of one of the small right triangles outlined in blue is \( A \) cm², while the area of the square outlined in red is \( B \) cm²". Let's count the number of blue triangles in the shaded region. Looking at the figure (a pinwheel-like shape), the shaded area: let's see, the red square is in the middle. Then, how many small triangles (area \( A \)) are there? Wait, maybe the square \( B \) is made of 4 triangles (so \( B = 4A \)), but the shaded area: let's count the triangles. Wait, the options: let's check the options. Let's assume that the square \( B \) is composed of 4 triangles (each \( A \)), so \( B = 4A \). Then, let's see the shaded area. Wait, maybe the shaded area has 8 triangles? No, wait, the options: \( 4A + B \), \( 8A + B \), \( 2A + B \), \( 1.5B \), \( 12A \). Let's check \( 1.5B \): if \( B = 4A \), then \( 1.5B = 6A \), no. Wait, maybe \( B = 8A \)? No. Wait, maybe the square \( B \) is made of 8 triangles? No, a square has 4 right triangles (when divided by two diagonals). Wait, maybe the figure has the square \( B \) and 4 triangles outside? No, the figure is a pinwheel with the square in the middle and 4 triangles attached? Wait, no, the figure shows a square (red) with diagonals, and 4 triangles outside? Wait, no, the shaded area: let's count the number of small triangles (area \( A \)) and the square \( B \). Let's suppose that the square \( B \) is composed of 4 triangles (each \( A \)), so \( B = 4A \). Then, the shaded area: how many triangles? Let's see, the options: \( 4A + B \): if \( B = 4A \), then \( 4A + 4A = 8A \), no. \( 8A + B \): if \( B = 4A \), then \( 8A + 4A = 12A \), which is an option. Wait, \( 12A \) is an option. Also, \( 1.5B \): if \( B = 8A \), then \( 1.5B = 12A \), which matches \( 12A \). Wait, let's think again. Let's count the number of triangles in the shaded area. Suppose the square \( B \) is made of 8 triangles? No, a square can't be made of 8 right triangles unless it's a square divided into 8 smaller triangles (like a square with midpoints of sides, dividing into 8 triangles). Wait, maybe the square \( B \) has area \( B = 8A \)? No, the problem says "small right triangle" is \( A \). Let's look at the options: \( 12A \) and \( 1.5B \) – if \( 12A = 1.5B \), then \( B = 8A \). Ah! So \( B = 8A \) (since \( 1.5 \times 8A = 12A \)). Then, let's check the other options. \( 4A + B = 4A + 8A = 12A \), no. \( 8A + B = 8A + 8A = 16A \), no. \( 2A + B = 2A + 8A = 10A \), no. Wait, no, maybe \( B = 8A \), and the shaded area is \( 12A \), which is \( 1.5B \) (since \( 1.5 \times 8A = 12A \)). Also, \( 4A + B \): if \( B = 8A \), then \( 4A + 8A = 12A \), yes! Wait, \( 4A + B \): if \( B = 8A \), then \( 4A + 8A = 12A \). Also, \( 1.5B \): if \( B = 8A \), \( 1.5 \times 8A = 12A \). Also, \( 12A \) is an option. Wait, let's verify:

- If \( B = 8A \) (so the square \( B \) is made of 8 small triangles, each area \( A \)), then:

- \( 4A + B = 4A + 8A = 12A \) – correct.

- \( 1.5B = 1.5 \times 8A = 12A \) – correct.

- \( 12A \) – correct.

Wait, but let's check the figure again. The figure is a pinwheel with a square in the middle (red) and 4 triangles outside? No, the figure shows a square with diagonals, and 4 triangles attached to the square's sides, and 4 triangles inside? Wait, maybe the total number of triangles in the shaded area: the square \( B \) has 8 triangles (each \( A \)), so \( B = 8A \). Then, the shaded area: the square \( B \) (8A) plus 4 triangles (4A) – total 12A. So \( 4A + B = 4A + 8A = 12A \), \( 1.5B = 1.5 \times 8A = 12A \), and \( 12A \) is also an option. Let's check the options:

- \( 4A + B \): if \( B = 8A \), then \( 4A + 8A = 12A \) – correct.

- \( 8A + B \): \( 8A + 8A = 16A \) – incorrect.

- \( 2A + B \): \( 2A + 8A = 10A \) – incorrect.

- \( 1.5B \): \( 1.5 \times 8A = 12A \) – correct.

- \( 12A \): correct.

Wait, but the problem says "check all that apply". So the correct options are \( 4A + B \), \( 1.5B \), and \( 12A \)? Wait, but let's re-examine. Maybe the square \( B \) is made of 4 triangles (each \( A \)), so \( B = 4A \). Then:

- \( 4A + B = 4A + 4A = 8A \) – no.

- \( 1.5B = 1.5 \times 4A = 6A \) – no.

- \( 12A \) – no.

So that can't be. So maybe \( B = 8A \). Let's assume that the square \( B \) is composed of 8 small right triangles (each area \( A \)), so \( B = 8A \). Then:

- \( 4A + B = 4A + 8A = 12A \) – correct.

- \( 1.5B = 1.5 \times 8A = 12A \) – correct.

- \( 12A \) – correct.

The other options: \( 8A + B = 16A \) (incorrect), \( 2A + B = 10A \) (incorrect). So the correct options are \( 4A + B \), \( 1.5B \), and \( 12A \). Wait, but let's check the figure again. The figure is a pinwheel with a square in the center (red) and 4 triangles attached to the square's sides, and 4 triangles inside the square? No, the square has diagonals, dividing it into 4 triangles, but maybe the square is actually made of 8 triangles? Wait, a square can be divided into 8 right triangles by drawing both diagonals and the midlines of the sides. Yes! If you draw both diagonals and the horizontal and vertical midlines, you get 8 small right triangles. So the square \( B \) is composed of 8 small triangles, each with area \( A \), so \( B = 8A \). Then, the shaded area: the square \( B \) (8A) plus 4 more triangles (4A) – total 12A. So:

- \( 4A + B = 4A + 8A = 12A \) – correct.

- \( 1.5B = 1.5 \times 8A = 12A \) – correct (since \( 1.5 \times 8A = 12A \)).

- \( 12A \) – correct.

The other options: \( 8A + B = 16A \) (too big), \( 2A + B = 10A \) (too small). So the correct options are \( 4A + B \), \( 1.5B \), and \( 12A \).

Step2: Verify each option

- Option 1: \( 4A + B \). If \( B = 8A \), then \( 4A + 8A = 12A \) – correct.

- Option 2: \( 8A + B \). \( 8A + 8A = 16A \) – incorrect.

- Option 3: \( 2A + B \). \( 2A + 8A = 10A \) – incorrect.

- Option 4: \( 1.5B \). \( 1.5 \times 8A = 12A \) – correct.

- Option 5: \( 12A \) – correct (matches the total).

Answer:

The correct options are:

• \( 4A + B \)
• \( 1.5B \)
• \( 12A \)